1
vote
1answer
29 views

Linear constraint in convex optimization

Is it true that the solution to a linearly constrained convex minimization problem can only be placed on the boundary of the constraint set, for any nonlinear convex objective, e.g. $$ \min_x f(x)$$ ...
1
vote
1answer
58 views

Convexity of a rational function

I am attempting to (dis)prove that the function $$\frac{4x+3y+2}{x^2+xy+2x+y}$$ is convex for $x,y>0$. Attempting to differentiate the function does not seem like a good idea (or am I making a ...
0
votes
1answer
27 views

Finding the maximum/minimum of a homogeneous function on $R^n$

Suppose that $f:R^n\to R$ is homogeneous. Also, suppose that the $argmin_xf(x)$ is non-empty. Is it true that if there exist $x^*\in R^n$ such that $f(x^*)=0$, then $x^*=argmin_xf(x)$?
1
vote
1answer
54 views

Partial derivative on convex set

If we have a function $f:U \rightarrow R$ ($U \subset R^n$) which is partially differentiable on a convex set U with $\frac{\delta f}{\delta x_1} = 0$ for all $x \in U$. How can we prove that $f$ ...
2
votes
0answers
31 views

Strictly convex self-concordant function

Some definitions: A function $f:R^n\rightarrow R$ is convex[strictly convex] if for every $\lambda\in[0,1]$ [$\lambda\in(0,1)$] and for every $x,y$ [$x\neq y$] in $R^n$ we have $f(\lambda ...
0
votes
2answers
49 views

Non-elementwise Matrix Derivatives

Let A,B,C,D,X be matrices. I'd like to perform a Gradient Descent minimization to the loss functin $$ tr[(AXBX^TC-D)^T(AXBX^TC-D)] $$ My question is, how to take the gradient efficiently w.r.t. $B$? ...
0
votes
1answer
55 views

Confusion over a proof that gradient is perpendicular to the level set

To prove that the vector $\nabla{f}(x_0)$ is orthogonal to the tangent vector to "an arbitrary smooth curve" passing through $x_0$ on the level set determined by $f(x)=f(x_0)$ the following proof is ...
0
votes
1answer
346 views

Pointwise supremum of a convex function collection

In Hoang Tuy, Convex Analysis and Global Optimization, Kluwer, pag. 46, I read: "A positive combination of finitely many proper convex functions on $R^n$ is convex. The upper envelope (pointwise ...
1
vote
1answer
107 views

Unboundedness of Quadratic Function

I was reading about Trust Region Methods for solving Nonlinear Optimization problems are came across this statement in my notes: If the quadratic approximation to the function $f(x)$ which is given ...
2
votes
2answers
109 views

Constrained maximization problem

I need help with the following optimization problem $$ \max\;\alpha\ln(x(1-y^2))+(1-\alpha)\ln(z) $$ where the maximization is with respect to $x,y,z$, subject to \begin{align} \alpha ...
2
votes
1answer
956 views

Prove the supremum of the set of affine functions is convex

Let $\langle f_i \rangle _{i \in I}$ be a family of affine functions on a convex and compact set $\Omega \subset \mathbb{R^d}$ such that $f_i = a_i.x +b_i$ for $x \in \Omega$. Prove that f, defined by ...
0
votes
1answer
253 views

Compact Set for Dummies

Can any one tell me in simple words what is a compact set? I read the definition of Compact set, but do not get it. BTW, I do not know topology. In particular, is the probability simplex, $W\ge0, ...
0
votes
1answer
267 views

Global Min-Max Optimization

When is \begin{equation} \min_X \max_Y f(X,Y) \end{equation} globally solvable? (i.e. we can find global solution for the optimization problem?) I am not looking for reformulations. Is it only when ...
3
votes
1answer
118 views

Lower bound of a function

Given $x \geq y > 0$ and an integer $n$. We want to minimize the following term $\sum_{i=1}^n (x_i^2 - x_iy_i)$ over $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ non-negative such that $\sum_{i=1}^n x_i ...
1
vote
0answers
164 views

Convexity of a Set

Consider the following function, $$ f(x, y) = e^{m e^{-y}+n e^{-x}-x-y} \left(a x e^y+b e^x y+c x y\right) $$ where $a, b, c, m$ and $n$ are positive constants. I want to show $f(x, y)$ is ...
1
vote
1answer
192 views

Supremum of vector dot-product

I am curious what is the precise math reasoning behind this: $$ \sup \{ a_i^T u | \lVert u\rVert_2 \leq r \} = r \lVert a_i \rVert_2$$ It is on page 148, last line, of Boyd's Convex Optimization ...
1
vote
2answers
338 views

Proving quadratic function is bounded (vector input)

I am reading Boyd's Convex Optimization textbook and I am looking at example 3.22, line 2. It says $y^T x - \frac12 x^T Q x$ is bounded from above for all possible values of $y$. Also, it is ...
0
votes
1answer
338 views

Derivative of a Scalar Function

In Boyd's Convex Optimization text, on page 86, there is the following equation: $$f(x) = h(g(x)) = h(g_1(x), \cdots, g_k(x))$$ Then, in order to show convexity, we take the second derivative, ...